Solution: 2015 Fall Final - 14
Author: Michiel SmidQuestion
Let $n$ be the number of students who are writing this exam. Each of these students has a uniformly
random birthday, which is independent of the birthdays of the other students. We ignore leap years;
thus, the year has 365 days. Define the event
- A = "at least one student's birthday is on December 21".
(a)
$1 - (1/365)^{n}$
(b)
$1 - (364/365)^{n}$
(c)
$n \cdot (1/365) \cdot (364/365)^{n-1}$
(d)
$365 \cdot n \cdot (364/365)^{n-1}$
Solution
Let B be the event that no student’s birthday is on December 21.
The first student has a birthday on any day except December 21: $ \frac{364}{365} $
The second student has a birthday on any day except December 21: $ \frac{364}{365} $
…
The $n$th student has a birthday on any day except December 21: $ \frac{364}{365} $
$ Pr(B) = {\left( \frac{364}{365} \right)}^{n} $
$ Pr(A) = 1 - Pr(B) $
$ Pr(A) = 1 - {\left( \frac{364}{365} \right)}^{n} $